\(\text{SL}_2\) action on the cohomology of a rank two Abelian group with arbitrary coefficient domain (Q2758965)

Let \(C_n\) denote the cyclic group of order \(n\), \(C_n^2=C_n\times C_n\), and \(R\) be a commutative ring. Of interest here are the cohomology groups \(H^m(C_n^2,R)\) for a natural number \(m\) with \(R\) considered as a trivial \(RC_n^2\)-module. For certain rings \(R\), these cohomology groups are known. For example, \textit{G. R. Chapman} [Proc. Lond. Math. Soc., III. Ser. 45, 564-576 (1982; Zbl 0565.20032)] determined the integral (and other) cohomology groups when \(n\) is an odd prime.NEWLINENEWLINENEWLINEThe authors' goal is to determine the structure for an arbitrary commutative ring \(R\) and moreover, not just the \(R\)-module structure, but also the structure as an \(\text{SL}_2(\mathbb{Z})\)-module. Here the action of \(\text{SL}_2(\mathbb{Z})\) is that induced from the natural action of \(\text{SL}_2(\mathbb{Z})\) on \(C_n^2\). Once the integral cohomology is known, the universial coefficient theorem can be used to obtain \(H^m(C_n^2,R)\) up to a short exact sequence. Using elementary computations with resolutions, the authors identify a short exact sequence \(0\to A\to H^m(C_n^2,R)\to B\to 0\) of \(\text{SL}_2(\mathbb{Z})\)-modules. The nature of the sequence depends on whether \(m\) is even or odd. For \(m\) even, precise conditions are identified under which the sequence splits. The splitting question is not answered for \(m\) odd. One is naturally led to ask whether such computations can be done for higher rank groups.